MinT - Best Known (190−69, 190, s)-Nets in Base 3

Best Known (190−69, 190, s)-Nets in Base 3

(190−69, 190, 156)-Net over F₃ — Constructive and digital

Digital (121, 190, 156)-net over F₃, using

8 times m-reduction [i] based on digital (121, 198, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 99, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(190−69, 190, 236)-Net over F₃ — Digital

Digital (121, 190, 236)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(190−69, 190, 3005)-Net in Base 3 — Upper bound on s

There is no (121, 190, 3006)-net in base 3, because

1 times m-reduction [i] would yield (121, 189, 3006)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 511414 139178 074071 282238 451146 356721 677408 957808 998253 513891 860878 104676 644215 049904 705725 > 3¹⁸⁹ [i]